Timeline for Let $X_t$ be an ARIMA(1,1,1) process and $Y_t = Y_{t-1} + X_t$. What kind of process is $Y_t$?
Current License: CC BY-SA 4.0
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| Nov 11, 2023 at 12:11 | comment | added | mlofton | I don't know if it's freely available ( I found it one of my boxes when I was looking for something else cause I moved ) but here's the interview. Definitely worth a read if you can get your hands on it. cambridge.org/core/journals/econometric-theory/article/abs/… | |
| Nov 11, 2023 at 12:09 | comment | added | mlofton | It's an interesting coincidence because, recently, I was reading a 2006 interview of Manfred Deistler ( top researcher dealing in systems theory which has a lot of overlap with econometerics and stats ) and he said that a colleague showed him that, empirically, the standard optimization algorithms for ARIMA estimation ( BFGS etc ) return the wrong ARIMA parameter estimates 70 percent of the time. He has developed improved methods ( coordinate methods and subspace approaches ) that improve on the convergence issue. The interview is in econometric theory 2006 if you're interested. | |
| Nov 11, 2023 at 12:03 | comment | added | mlofton | I agree that the random walk needs an initial value and so will the $Z_t$. But there are "methods" ( I don't know how well they work ) that generate a pseudo initial value. All I'm saying is that I don't think estimating the parameters of the model has any unusual problems that a standard ARIMA estimation problem doesn't have. | |
| Nov 11, 2023 at 7:52 | comment | added | Sextus Empiricus | And besides using least squares, the parameters can also be 'solved'/found by differencing twice and finding the parameters for the ARMA(1,1) process (as you mentioned in the first comment). I have removed the 'has no solution' part which has been confusing. You do have a solution for finding the parameters in the problem... but, it is in finding the likelihood function that there is no solution. | |
| Nov 11, 2023 at 7:44 | history | edited | Sextus Empiricus | CC BY-SA 4.0 |
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| Nov 11, 2023 at 7:43 | comment | added | Sextus Empiricus | @mlofton If I have a random walk with the given recursive formula $X_t = X_{t-1} + \epsilon$ then, conditional on the parameters, you do not have a unique distribution for the vector $\mathbf{X}$ (you would need some anchor point as well, e.g. given that $X_0 = 0$, you can describe the distribution of $X_t$ for $t \geq 0$). I am not sure, but I believe that you can always (without problems) solve the parameters for the recursive formula by using a least squares formulation. But it is not possible to solve the equations with maximum likelihood / method of moments. | |
| Nov 11, 2023 at 2:19 | comment | added | mlofton | Right. It's not. But as far as estimation, can one not take $Z_t$ and estimate the parameters of the ARMA(1,1) process ? Maybe I didn't understand your comment about stable process and no solution. Thanks. | |
| Nov 10, 2023 at 17:30 | comment | added | Sextus Empiricus | @mlofton sure, taking twice the difference $Z_t := (Y_{t} - Y_{t-1}) - (Y_{t-1} - Y_{t-2})$ is an ARMA(1,1) process, but $Y_t$ is not that ARMA(1,1) process in the same way as a random walk is not equivalent to Gaussian white noise (instead it is the sum of that noise). | |
| Nov 10, 2023 at 17:19 | comment | added | mlofton | Why can the OP not difference the series twice and then estimate an ARMA(1,1) on that series ? Where does the instability and lack of solution arise ? Thanks. | |
| Nov 10, 2023 at 16:53 | history | answered | Sextus Empiricus | CC BY-SA 4.0 |